## Statistics- The interval is called a confidence interval and the spread

The interval is called a

confidence interval and the spread of the interval is dependent in part on the

spread of the sample statistic around the true population parameter. We know

how sample proportions vary around the true population proportion from last

lesson when we introduced the sampling distribution of a sample proportion. The

spread of our sampling distribution depended on the population proportion value

(p) and the sample size (n) (i.e. the standard deviation of the sampling

distribution was derived using a formula that used the population proportion

and sample size). However, now we do not know the population proportion (if we

did, then we would not have to use our sample value to make an inference about

the population value). We substitute the sample proportion value into our

formula for the standard deviation of the sampling distribution and now call

this measure of spread the standard error of the sample proportion.

Standard Error of the Sample

Proportion (SEP) = estimated standard deviation of the sample proportion

sampling distribution = √(sample proportion*(1-sample proportion)/sample size)

1. A random sample of 100

freshmen students at University Park campus is taken to estimate the percentage

of freshmen that have taken a tour of Pattee Library during orientation

week. Of the 100 freshmen, 40 (40%) of

them have taken a tour of Pattee Library.

The standard error (SE) for a sample percentage in this situation is

estimated to be approximately:

A) 4.0%

B) 4.9%

C) 0.24%

D) 24%

2. A random sample of 400

freshmen students at University Park campus is taken to estimate the percentage

of freshmen who have taken a tour of Pattee Library during orientation

week. Of the 400 freshmen, 160 (40%) of

them have taken a tour of Pattee Library.

The standard error (SE) for a sample percentage in this situation is

estimated to be approximately:

A) 4.9%

B) 4.0%

C) 2.45%

D) 24%

3. When the sample size

increases by 4 times (going from a sample of size 100 to a sample of size 400

with everything else remaining the same), the SE of the sampling distribution

for our sample proportion is:

A) Â½ as large as it was before

the sample size increase

B) 2 times larger than it was

before the sample size increase

C) 4 times larger than it was

before the sample size increase

D) Â¼ as large as it was before

the sample size increase

4. As sample size increases:

A) the SE increases

B) the SE remains the same

C) the SE decreases

Use the information below to

answer the questions that follow:

The confidence interval for

our true population proportion (for large random samples) is given by:

sample proportion +/- z*SEP

Where z* is a multiplier of

the SEP and the z value used depends on the level of confidence (see Table 10.1

in the online notes for z values for a particular level of confidence).

5. A random sample of 400

freshmen students at University Park campus is taken to estimate the percentage

of freshmen who have taken a tour of Pattee Library during orientation

week. Of the 400 freshmen, 160 (40%) of

them have taken a tour of Pattee Library.

The 95% confidence interval for the population percentage for all

freshmen who have taken a tour of Pattee Library during orientation week is:

A) 40% +/- 2.45%

B) 40% +/- 4.90%

C) 95% +/- 40%

D) 5% +/- 40%

6. An exit poll was done on

election night. The exit poll found that

204 out of 400 randomly selected voters (51%) had voted for the incumbent

Senator. The 95% confidence interval

for the population percentage of all voters who voted for the incumbent Senator

in that election is:

A) 51% +/- 5%

B) 51% +/- 2.5%

C) 95% +/- 5%

D) 95% +/- 2.5%

7. A random sample of 225

residents of a small mid-western town found that 80% approve of a new parking

ordinance. The 95% confidence interval

for the true population percentage of all residents who favor the parking

ordinance is:

A) 80% +/- 5.33%

B) 80% +/- 2.666%

C) 80% +/- 8%

D) 80% +/- 2%

8. Based on an exit poll done

on election night, the 95% confidence interval for the population percentage of

all voters who voted for the incumbent Senator was 52% +/- 3%. Based on this confidence interval, can we

state that we are 95% confident that the Senator got a majority of the vote

(greater than 50%) and was re-elected?

A) Yes

B) No

Use the information below to

answer the questions that follow:

The interval is called a

confidence interval and the spread of the interval is dependent in part on the

spread of the sample statistic around the true population parameter. We know

how sample means vary around the true population mean from last lesson when we

introduced the sampling distribution of a sample mean. The spread of our

sampling distribution depended on the population standard deviation value and

the sample size (i.e. the standard deviation of the sampling distribution was

derived using a formula that used the population standard deviation and sample

size). However, now we do not know the population value for our calculation. We

substitute the sample standard deviation value into our formula for the

standard deviation of the sampling distribution and now call this measure of

spread the standard error of the sample mean.

Standard Error of the Sample

Mean (SEM) = estimated standard deviation of the sample mean sampling

distribution = sample standard deviation/√n

9. A random sample of 81

residents of a Central Pennsylvania county found that the average restaurant

expenditure in the last month was $60 with a sample standard deviation of

$18. The standard error (SE) for the

average amount spent will be:

A) $18

B) $0.22

C) $2

D) $6.67

10. A random sample of 100

statistics students finds that the average time spent on an exam is 50 minutes

with a sample standard deviation of 5 minutes.

The standard error (SE) for the average time spent is:

A) 10.0 minutes

B) 1.0 minutes

C) 5.0 minutes

D) 0.5 minutes

11. A random sample of 400

statistics students finds that the average time spent on an exam is 50 minutes

with a sample standard deviation of 5 minutes.

The standard error (SE) for the average time spent is:

A) 0.5 minutes

B) 0.25 minutes

C) 0.0125 minutes

D) 5.0 minutes

12. Everything else being

equal, an increase in sample size from 100 to 400 will result in the standard

error (SE) for the average _________.

A) decreasing to Â¼ of what it

was for the sample size of 100

B) decreasing to Â½ of what it

was for the sample size of 100

C) increasing to 2 times what

it was for the sample size of 100

D) increasing to 4 times what

it was for the sample size of 100

Use the information below to answer

the questions that follow:

The confidence interval for

our true population mean (for large random samples, n > 30) is given by:

sample mean +/- z*SEM

Where z* is a multiplier of

the SEM that depends on the level of confidence (see Table 10.1 in the online

notes for z values for a particular level of confidence).

13. A random sample of 100

full time graduate students in the College of Science is taken to estimate the

amount spent on textbooks this semester.

The sampled students spent an average of $400 with a sample standard

deviation of $100. A 95% confidence

interval for the average amount spent by all full time graduate students in the

College of Science would be:

A) $400 +/- $100

B) $400 +/- $200

C) $400 +/- $10

D) $400 +/- $20

14. A random sample of 225

Penn State football season ticket holders is taken to estimate the population

average amount spent in State College on a typical home football game

weekend. The sample average was $500

with a sample standard deviation of $75.

A 95% confidence interval for the average amount spent by all Penn State

Football season ticket holders on a typical home football game weekend is:

A) $500 +/- $75

B) $500 +/- $5

C) $500 +/- $10

D) $500 +/- $150

15. A random sample of 81

Statistics undergraduate students found that the average amount of hours of

sleep per week during the past semester was 45.5 hours with a sample standard

deviation of 3 hours. A 95% confidence

interval for the average amount of hours of sleep per week by all Statistics

undergraduate students is:

A) 45.5 +/- 0.666 hours

B) 45.5 +/- 0.333 hours

C) 45.5 +/- 3 hours

D) 45.5 +/- 6 hours

16. A 95% confidence interval

for the average amount spent on textbooks by undergraduate students at a large

state university per semester is $400 +/- $50. Would it be correct to state

that the true population average could be $360?

A) Yes

B) No

We can also create a

confidence interval for the difference between two population proportions or

population means.

If we were to randomly choose

a sample from two different groups (independent samples) and then compare those

two sample values to one another and then do this many times with different

samples from each group, we would end up with a sampling distribution of the

DIFFERENCE between two sample values. This sampling distribution has a measure

of spread that is estimated by:

standard error of the

difference between two sample statistics (SED): = √((standard error of first

sample)^2 + (standard error of the second sample)^2)

We do not use the SED formula

when we have matched pairs (dependent samples). See example 10.7 in the online

notes.

17. A consumer testing

organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of PC

(independent samples). The 49 PCs in

group 1 have an average battery life of 11 hours with a sample standard

deviation of 0.7 hours. The 49 PCs in

group 2 have an average battery life of 9 hours with a sample standard

deviation of 1.4 hours

The standard error for the

difference in the two averages is:

A) 0.7 + 1.4 = 2.1 hours

B) √(0.7^2 + 1.4^2) = 1.56

hours

C) √(0.1^2 + 0.2^2) = 0.223

hours

D) 0.1 + 0.2 = 0.3 hours

The confidence interval for

our true population difference is given by:

sample difference +/- z*SED

Where z* is a multiplier of

the SED that depends on the level of confidence (see Table 10.1 in the online

notes for z values for a particular level of confidence). When making

conclusions about our population values using a confidence interval of the

difference, we look to see if there is a â0â within the interval. If there is a

â0â within the interval, we cannot conclude that there is a difference between

the two population values.

18. A consumer testing

organization is comparing the battery life of two different models of PCs. They randomly sample 49 of each model of

PC. The 49 PCs in group 1 have an

average battery life of 11 hours with a sample standard deviation of 0.7

hours. The 49 PCs in group 2 have an

average battery life of 9 hours with a sample standard deviation of 1.4 hours.

The 95% confidence interval of

the difference between the two group population means is:

A) (11-9) +/- 2 * 0.223 = 2

+/- 0.446 hours

B) (11-9) +/- 1 * 0.223 = 2

+/- 0.223 hours

C) (11-9) +/- 2 * 1.56 = 2 +/-

3.12 hours

D) (11-9) +/- 3 * 0.223 = 2

+/- 0.669 hours

19. As stated in the online

notes, holding everything else equal, a four fold increase in sample size alone

(as when the sample size increases from 100 to 400) will cut the margin of

error in half. Remembering that the margin of error = z*SE (the margin of error

is the distance on either side of the sample value that our confidence interval

covers), what is the reason that the margin of error is cut in half?

20. What happens to the width

of a confidence interval when the confidence level is increased from 90% to 95%

(holding all else constant)? Why? (Provide the reasoning behind your answer.)

21. We randomly select 1000

adults from a population of 2 million and also randomly select 1000 adults from

a population of 20 million. If the sample standard deviations are the same, how

will the margins of error compare for 95% confidence intervals for the true

population average? Will the margin of error for the sample from the larger

population be greater than, the same, or less than the margin of error for the

sample from the smaller population? Why? (Provide the reasoning behind your

answer.)

22. We have calculated a 95%

confidence interval for the difference between two population proportions. The

interval is 4% +/- 5%. Can we conclude that there is a difference between the

two group population proportions? Yes or No? Why? (Provide the reasoning behind

your answer.)

23. A group of college age

males and a group of college age females were sampled. and 95% confidence

intervals were created for the true population average of the amount spent on

pizza and related items in the past month. The male confidence interval is from

$40 to $60 ($50 +/- $10). The female confidence interval is from $25 to $35

($30 +/- $5). Is there a difference between the two group population mean

amounts spent? Why? Provide the reasoning behind your answer. (HINT: See page

10.4 in the online notes: “Statistical Significance and Confidence

Intervals.”)

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